The Hodge star mean curvature flow on a 3‑dimensional Riemannian or pseudo-Riemannian manifold is one of nonlinear dispersive curve flows in geometric analysis. Such a curve flow is integrable as its local differential invariants of a solution to the curve flow evolve according to a soliton equation. In this paper, we show that these flows on a 3‑sphere and 3‑dimensional hyperbolic space are integrable, and describe algebraically explicit solutions to such curve flows. Solutions to the (periodic) Cauchy problems of such curve flows on a 3‑sphere and 3‑dimensional hyperbolic space and its Bäcklund transformations follow from this construction.

中文翻译：

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3球和双曲3空间上的星形平均曲率流

三维黎曼流形或拟黎曼流形上的霍奇星平均曲率流是几何分析中的非线性色散曲线流之一。这样的曲线流是可积分的，因为其对孤流的解的局部微分不变量根据孤子方程发展。在本文中，我们证明了这些在3维球面和3维双曲空间上的流是可积的，并描述了这种曲线流的代数显式解。这种曲线在（3维球面和3维双曲空间上）的（周期）柯西问题的解决方案及其构造的Bäcklund变换。